Laplace transform

Definition

The Laplace transform is an integral transform that converts a function of time $f(t)$ into a function of a complex variable $s$, denoted as $F(s)$. It is widely used to solve differential equations by transforming them into algebraic equations.

5 Must Know Facts For Your Next Test

1. The Laplace transform of a function $f(t)$ is defined as $F(s) = \int_0^\infty f(t)e^{-st}dt$.
2. It can simplify the process of solving linear differential equations with given initial conditions.
3. Common properties include linearity, differentiation in the time domain, and integration in the time domain.
4. For improper integrals, the convergence of the Laplace transform depends on the behavior of $f(t)$ as $t \to \infty$.
5. $\mathcal{L}\{1\} = \frac{1}{s}$ for $Re(s) > 0$.

Review Questions

• What is the formula for the Laplace transform of a function $f(t)$?
• How does the Laplace transform help in solving differential equations?
• What condition must be met for an improper integral to converge when applying the Laplace transform?

Related terms

Integral Transform: A mathematical operation that converts a function into another function via integration.

Improper Integral:

An integral where either the interval of integration is infinite or the integrand has an infinite discontinuity.

\(e^{-st}\): The exponential decay factor used in the definition of the Laplace transform.